Marginal Sensitivity Models for Weighted Estimators

Why Bound the Propensity Score?

Inverse-probability weighting uses treatment probabilities to construct a pseudo-population in which measured covariates are balanced across treatment groups. In an observational study, however, the propensity score estimated from recorded covariates may not capture all factors that influenced treatment. An unmeasured covariate can make treatment more or less likely for otherwise similar individuals, changing the weights and potentially shifting the estimated causal effect.

A marginal sensitivity model makes this concern quantitative. Rather than asserting that hidden confounding is absent, it limits how far the treatment odds conditional on both measured and unmeasured covariates may depart from the propensity score conditional on measured covariates alone. The sensitivity parameter describes the permitted departure; it does not identify the true amount of confounding.

Definition: Let $A\in\{0,1\}$ denote treatment, $X$ the measured covariates, and $U$ unmeasured covariates. Define the marginal propensity score $e(X)=P(A=1\mid X)$ and the latent conditional propensity score $q(X,U)=P(A=1\mid X,U)$. A marginal sensitivity model with parameter $\Gamma\geq 1$ requires, for individuals in the covariate support, $$ \frac{1}{\Gamma} \leq \frac{q(X,U)/(1-q(X,U))}{e(X)/(1-e(X))} \leq \Gamma. $$

The restriction compares odds, not probabilities. At $\Gamma=1$, it requires $q(X,U)=e(X)$: given $X$, the unmeasured covariates do not change treatment odds. Larger values allow more departure. For example, $\Gamma=2$ permits the conditional treatment odds for a given $(X,U)$ to be as small as half, or as large as twice, the odds based on $X$ alone.

Because $e(X)$ is the treatment probability averaged over the distribution of $U$ given $X$, a compatible model also obeys $E[q(X,U)\mid X]=e(X)$. The odds restriction is not a claim that each individual's conditional propensity equals the marginal one. It bounds the individual-level variation while retaining the marginal propensity score as the average within each measured-covariate stratum.

From Odds Restrictions to Propensity Limits

For a fixed value $e=e(X)$, the odds restriction can be written as a range for the latent propensity $q=q(X,U)$. This conversion is useful because inverse-probability weights contain $q$ and $1-q$, rather than the odds themselves.

\frac{e}{e+\Gamma(1-e)} \leq q \leq \frac{\Gamma e}{1-e+\Gamma e}.

To obtain the lower limit, the smallest allowed odds are $e/\{\Gamma(1-e)\}$. If $q/(1-q)=r$, then $q=r/(1+r)$; substituting that smallest odds value gives $q=e/\{e+\Gamma(1-e)\}$. The same conversion using the largest allowed odds, $\Gamma e/(1-e)$, gives the upper limit. When $e$ is strictly between zero and one and $\Gamma$ is finite, both limits are also strictly between zero and one.

Worked Example: Translating an Odds Bound into Propensity Limits

Suppose an individual has measured-covariate propensity score $e(X)=0.20$, and the chosen sensitivity parameter is $\Gamma=2$. Find the permitted range for $q(X,U)$.

The lower limit is $$ \frac{0.20}{0.20+2(1-0.20)} = \frac{0.20}{1.80} = 0.1111. $$ The upper limit is $$ \frac{2(0.20)}{1-0.20+2(0.20)} = \frac{0.40}{1.20} = 0.3333. $$ Thus the model permits latent propensities from about 0.111 to 0.333 for this $X$, subject also to the requirement that their conditional average over $U$ is 0.20.

The corresponding marginal odds are $0.20/0.80=0.25$. At the lower limit, the odds are $0.1111/0.8889\approx0.125$, half the marginal odds. At the upper limit, they are $0.3333/0.6667=0.50$, twice the marginal odds. The probability limits are therefore consistent with the specified odds restriction.

Weighted Estimands Under the Model

Let $Y(1)$ and $Y(0)$ be the potential outcomes under treatment and control, and let $Y=A Y(1)+(1-A)Y(0)$ be the observed outcome under consistency. The target here is the average treatment effect, $\tau=E[Y(1)-Y(0)]$, in the population represented by the data.

To use the weighted representation below, the causal assumptions must be stated. Assume consistency, no interference between individuals, conditional exchangeability given $(X,U)$, and positivity: $Y(a)$ is independent of $A$ conditional on $(X,U)$, and $0<q(X,U)<1$ wherever the target population has support. Also assume the observed-data propensity $e(X)$ is correctly specified or consistently estimated. The sensitivity model adds the odds restriction above; it does not replace conditional exchangeability given $(X,U)$.

Result: Under these assumptions, for the true latent propensity $q(X,U)$, the average treatment effect has the inverse-probability-weighted representation $$ \tau = E\left[ \frac{A Y}{q(X,U)} - \frac{(1-A)Y}{1-q(X,U)} \right]. $$

The representation follows by conditioning on $(X,U)$. Conditional exchangeability and consistency imply $E[AY/q\mid X,U]=E[Y(1)\mid X,U]$, since $P(A=1\mid X,U)=q$. Similarly, $E[(1-A)Y/(1-q)\mid X,U]=E[Y(0)\mid X,U]$. Subtracting these conditional expectations and averaging gives $E[Y(1)-Y(0)]$.

The latent propensity is not observed, so the weighted representation cannot simply be evaluated at a known $q$. Instead, the model defines a collection of full-data explanations compatible with the observed data and with the odds restriction. Each explanation has a possible ATE; the lower and upper endpoints are the smallest and largest ATEs among those explanations.

Definition: Let $\mathcal{P}_{\Gamma}$ denote the set of full-data distributions that (1) agree with the observed distribution of $(X,A,Y)$, (2) satisfy consistency, no interference, conditional exchangeability given $(X,U)$, and positivity, and (3) obey the marginal sensitivity model with parameter $\Gamma$. The causal estimand bounds are $$ \tau_L(\Gamma) = \inf_{P\in\mathcal{P}_{\Gamma}} E_P[Y(1)-Y(0)], \qquad \tau_U(\Gamma) = \sup_{P\in\mathcal{P}_{\Gamma}} E_P[Y(1)-Y(0)]. $$

Equivalently, the optimization ranges over compatible latent propensities and latent covariate distributions, evaluating the weighted representation under each compatible explanation. The interval $[\tau_L(\Gamma),\tau_U(\Gamma)]$ is an identification region for the ATE under the stated model, not automatically a confidence interval. Calculating sharp endpoints can require an optimization that respects the observed data and the constraints on latent propensities; the propensity limits alone are not, in general, enough to read off the endpoints.

Worked Example: How Propensity Departures Change Weighted Contributions

Suppose $e(X)=0.40$ for an individual and $\Gamma=2$. The model allows $$ q_{\min}=\frac{0.40}{0.40+2(0.60)}=0.2500, \qquad q_{\max}=\frac{2(0.40)}{0.60+2(0.40)}=0.5714. $$ Consider the lower endpoint $q=0.25$. Relative to weighting with the marginal score, the treated contribution changes by the factor $$ \frac{e(X)}{q}=\frac{0.40}{0.25}=1.60. $$ The control contribution changes by $$ \frac{1-e(X)}{1-q}=\frac{0.60}{0.75}=0.8000. $$

For example, if this individual is treated and has outcome $Y=10$, the marginal-score term is $AY/e(X)=10/0.40=25$, while the latent-propensity term is $AY/q=10/0.25=40$. If instead the individual is a control with $Y=6$, the terms are $6/0.60=10$ and $6/0.75=8$, respectively. These are weighted contributions, not individual causal effects; the population effect requires averaging over individuals and comparing the treated and control contributions. In particular, one cannot choose $q$ independently for each contribution and call the resulting difference a sharp bound without enforcing compatibility with the observed distribution.

A Small Population Illustration

A simple hypothetical population shows why an unmeasured covariate can alter weighted estimates even when the sensitivity model permits only a finite departure. Let $X$ be constant, and let $U$ take two values with probability $0.5$ each. Assume the following treatment probabilities and potential outcomes.

Latent group $U$	Population proportion	$q(X,U)$	$Y(0)$	$Y(1)$
Low	0.50	0.25	2	8
High	0.50	0.75	6	12

Worked Example: Comparing Marginal and Latent-Propensity Weights

The marginal propensity is $e(X)=0.5(0.25)+0.5(0.75)=0.50$. Its odds are 1. The low group's odds are $0.25/0.75=1/3$, and the high group's are $0.75/0.25=3$. Their odds ratios relative to the marginal odds are $1/3$ and 3, so this population satisfies the model with $\Gamma=3$.

The average treatment effect in this population is $$ \tau = 0.50(8-2)+0.50(12-6) = 0.50(6)+0.50(6) = 6. $$ The marginal-score weighted treated mean contribution is $$ E\left[\frac{AY}{e(X)}\right] = \frac{0.50(0.25)(8)+0.50(0.75)(12)}{0.50} = \frac{5.50}{0.50} = 11. $$ The corresponding control contribution is $$ E\left[\frac{(1-A)Y}{1-e(X)}\right] = \frac{0.50(0.75)(2)+0.50(0.25)(6)}{0.50} = \frac{1.50}{0.50} = 3. $$ Their difference is $11-3=8$, which differs from the population ATE of 6.

Using the latent propensities instead gives $$ E\left[\frac{AY}{q(X,U)}\right] = 0.50(8)+0.50(12)=10, \qquad E\left[\frac{(1-A)Y}{1-q(X,U)}\right] = 0.50(2)+0.50(6)=4. $$ The difference is $10-4=6$, matching the ATE. This example illustrates a compatible explanation within the $\Gamma=3$ model; it does not establish that 6 is an endpoint of the bounds for some separate observed data set.

Interpreting the Sensitivity Parameter

The sensitivity parameter is best understood as a restriction on the strength of hidden selection into treatment, conditional on measured covariates. It is not the prevalence of an unmeasured confounder, the size of its outcome effect, or a direct measure of bias in outcome units. The same value of $\Gamma$ can lead to different ATE bounds in different studies because the observed covariates, outcomes, propensity scores, and compatible latent structures differ.

The bound sets are nested: if $\Gamma_2\geq\Gamma_1\geq1$, every distribution allowed at $\Gamma_1$ is also allowed at $\Gamma_2$. Therefore the lower endpoint cannot increase and the upper endpoint cannot decrease as $\Gamma$ grows. At $\Gamma=1$, the model rules out additional treatment-propensity variation due to $U$. If conditional exchangeability given $X$ is then appropriate, the region collapses to the usual identified ATE. For $\Gamma>1$, a wider region reflects the greater range of hidden-confounding structures admitted by the model.

This sensitivity analysis does not by itself account for sampling uncertainty. A sample estimate of the endpoints varies across samples, and uncertainty procedures must address that variation as well as estimation of $e(X)$. Nor does a narrow region prove that the model is correct: it may be narrow because the observed data constrain the compatible explanations, while the assumptions about exchangeability and the form of the sensitivity restriction remain substantive.

Common Pitfall

Wrong reasoning: “The model says the hidden covariate can at most double treatment odds, so the ATE estimate can be at most twice as large.”
Correction: $\Gamma$ bounds a ratio of treatment odds, not a ratio of causal effects or of bias. The effect bounds depend on how admissible propensity variation can reweight the observed outcomes, along with the constraints imposed by the observed data. They must be obtained from the estimand optimization, not by multiplying an estimate by $\Gamma$.

A related mistake is to treat the propensity limits for each $X$ as independent choices for every observed person. The latent propensities must be compatible with the marginal propensity and the observed data. Ignoring that compatibility can produce bounds that are not valid for the stated model.

Check Your Understanding

Use the definitions and assumptions in this tutorial to answer the following questions.

In a marginal sensitivity model, what two treatment odds are compared, and what does $\Gamma=1$ imply?
For $e(X)=0.30$ and $\Gamma=2$, calculate the lower and upper permitted values of $q(X,U)$.
Why does the weighted representation use $q(X,U)$ rather than $e(X)$ when conditional exchangeability is assumed given $(X,U)$?
What additional compatibility condition connects the latent propensity to the marginal propensity within levels of $X$?
Explain why $[\tau_L(\Gamma),\tau_U(\Gamma)]$ is an identification region rather than automatically a confidence interval.